Two-Fold Homotopy of 2-Crossed Module Maps of Commutative Algebras

TL;DR

This paper develops a homotopy theory framework for 2-crossed modules of commutative algebras, introducing 2-fold homotopies and establishing a 2-groupoid structure under certain conditions.

Contribution

It defines 2-fold homotopies between 1-fold homotopies of 2-crossed module maps and proves the existence of a 2-groupoid structure when the domain is free up to order one.

Findings

Defined 2-fold homotopy between 1-fold homotopies.

Established a 2-groupoid structure for certain 2-crossed modules.

Proved the structure exists when the domain algebra is polynomial.

Abstract

We address the homotopy theory of 2-crossed modules of commutative algebras. In particular, we define the concept of a 2-fold homotopy between a pair of 1-fold homotopies connecting 2-crossed module maps $\A \to \A'$. We also prove that if the domain 2-crossed module $\A$ is free up to order one (i.e. if the bottom algebra is a polynomial algebra) then we have a 2-groupoid of 2-crossed module maps $\A \to \A'$ and their homotopies and 2-fold homotopies.